Prajakta Nimbhorkar

The Planar k-Means Problem is NP-Hard

In the k-means problem, we are given a finite set S of points in

The planar k-means problem is NP-hard

\Re^m

Pseudorandom generators for group products: extended abstract

, and integer k ≥ 1, and we want to find k points (centers) so as to minimize the sum of the square of the Euclidean distance of each point in S to its nearest center. We show that this well-known problem is NP-hard even for instances in the plane, answering an open question posed by Dasgupta [6].

Graph Isomorphism for K_{3, 3}-free and K_5-free graphs is in Log-space.

In the k-means problem, we are given a finite set S of points in @?^m, and integer k>=1, and we want to find k points (centers) so as to minimize the sum of the square of the Euclidean distance of each point in S to its nearest center. We show that this well-known problem is NP-hard even for instances in the plane, answering an open question posed by Dasgupta (2007) [7].

3-connected Planar Graph Isomorphism is in Log-space

We prove that the pseudorandom generator introduced by Impagliazzo et al. (1994) with proper choice of parameters fools group products of a given finite group G. The seed length is O((|G|O(1) + log 1/δ)log n), where n is the length of the word and δ is the allowed error. The result implies that the pseudorandom generator with seed length O((2O(w log w) + log 1/δ)log n) fools read-once permutation branching programs of width w. As an application of the pseudorandom generator one obtains small-bias spaces for products over all finite groups Meka and Zuckerman (2009).

Rank-Maximal Matchings – Structure and Algorithms

Graph isomorphism is an important and widely studied computational problem with a yet unsettled complexity. However, the exact complexity is known for isomorphism of various classes of graphs. Recently, \cite{DLNTW09} proved that planar isomorphism is complete for log-space. We extend this result %of \cite{DLNTW09} further to the classes of graphs which exclude

Log-space Algorithms for Paths and Matchings in k-trees

K_{3,3}

Popularity at Minimum Cost

or

Dynamic Rank-Maximal Matchings

K_5

How Good Are Popular Matchings

as a minor, and give a log-space algorithm. Our algorithm decomposes